I have two different account on AnHosting. I need to move my blog on one account to my new account on the same webhost, will this affect page rank or my Google search ranking?
No it wont. But make sure you transfer the blog without resulting any error on the new server such as permalink setting, re-write url setting etc.
I moved my site to a new server within the same hosting company but on a new IP as we were running out of resources on the old machine in the beginning of December. Just a couple of days after the move we dropped a full page and haven't moved since. I know people say moving servers doesn't affect rank but every time I've ever moved a site to a new IP it's dropped.
it won't affect your ranking. just make sure your website is already setup on the new server before change the nameserver to avoid downtime.
the links that are pointing to your site will still be pointing to your site so it shouldn't affect it.